The answer consists of two parts. The first answer is that al-Rahman is a general attribute and al-Rahim is a specific attribute. Al-Rahman is general for everyone, while al-Rahim is specific to the believers, as Allah the Exalted said:

The Shaykh : The answer consists of two parts. The first answer is that al-Rahman is a general attribute and al-Rahim is a specific attribute. Al-Rahman is general for everyone, while al-Rahim is specific to the believers, as Allah the Exalted said: The second meaning is that al-Rahman is with regard to the attribute, and al-Rahim is with regard to the action. His attribute is mercy, and therefore it came in the form of fa'lan, which indicates abundance and fullness. For example, ghadiban is for one who is full of anger, sakran is for one who is full of intoxication, and rayyan is for one whose belly is full of water. So when the attribute was intended, it came in the pattern of fa'lan. But when the action was intended, it came in the name al-Rahim. The second is closer: that al-Rahman is with regard to the attribute, and al-Rahim is with regard to the action, which is conveying mercy to the one who is shown mercy.